Steve S. answered • 01/04/14

Tutoring in Precalculus, Trig, and Differential Calculus

C_1 / D_1 = C_2 / D_2 = C_3 / D_3 = C_4 / D_4 = ... = C_n / D_n := π or pi.

As indicated, this common ratio is defined (:=) as the number π or pi. Note that even though pi is defined as a ratio of numbers it is NOT rational because it is not a ratio of integers; either C_i or D_i MUST be irrational.

A circle can be approximated with a regular polygon with n sides of length s and interior radius r (shortest distance between center and one side). The area of such a regular polygon is (just add up all the triangles):

Area_regPoly = (1/2)nsr.

The Perimeter of the polygon is P = ns.

As n increases to very large numbers, the Regular Polygon looks more and more like a Circle. The Perimeter becomes the Circumference and the Area becomes:

Area_Cirle = (1/2)*C*r = (1/2)*(C/D)*r*D = (1/2)*(pi)*r*(2r)

Area_Cirle = (pi)*r^2 = πr^2

=====

A car wheel with Diameter, D = 77 cm = 0.77 m, has a Circumference, C = π D = π * 0.77 m.

To cover a distance of 72.6 m, the wheel will have to rotate 72.6 m / (π * 0.77 m) ≈ 30.01207498304312 ≈ 30.0 times.